4 Answers
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The idea that God cannot be in the presence of sin may come from the prayer in Habakkuk 1:13. In the NIV it says, "Your eyes are too pure to look upon evil."

The question, then, is, "Is this teaching the God is unable to look at evil?" If it is teaching that, then this would need to fit with the whole of Scripture.

Psalm 139 expressly states that God knows and sees everything that occurs, which would include evil. God also came to Adam and Eve after their sin. He visited Sodom to see the evil that was there. He observed the Israelites fashioning a golden calf and worshiping it. In Job, Satan comes into God's presence and has discussions with Him. Jesus, too, walked among us and certainly was in the presence of evil His whole life.

So, it seems clear that God does see evil and He does allow sin to come into His presence. Indeed, if all have sinned (and continue to do so), God must be able to be in the presence of sin to indwell those who have placed their trust in the redemption of Christ.

A key point here is that those who have trusted Christ are, according to the Scriptures, declared righteous in His sight--forgiven. The debt is fully paid. No charge can be entertained against us.

Now God's work of redemption will be fully complete when He fully removed that which He hates (sin) from that which He loves (His people).

In Job 1 Satan who is the father of sin goes into the presence of God and speaks to him directly. So to say that God can not be in the presence of sin is wrong. The main reason you are asking this question is because you are struggling with the idea that the trinity is real. lets look at Matthew 28:19

Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.

Now if the Holy Spirit is not one of the three persons of God then we would not be asked by Jesus Himself make disciples or baptize them in His name. He (the Holy Spirit) is clearly mentioned as a part of the three persons of God.

God can dwell in a sinner, provided that they have accepted Christ as their personal savior, for he removes us from our sin "As far as the east is from the west, so far He removed our transgressions from us” (Psalm 103:12)
"He covers our sins, blots them out as a thick cloud" (Rom. 4:7; Isa. 43:25; 44:22).

But let's understand the Trinity a little better. If you may, The Father is not the Son. The Son is not the Spirit. The Spirit is not the Father. Yet they are one. Our God. I emphasize our, because it is important to keep in mind who.

Our God (the Father) cannot be in the presence of sin, usually comes from Isaiah 59:2

But your iniquities have separated you from your God; your sins have
hidden his face from you, so that he will not hear

This separation is kinda why The Son (our God) came to save us from sin in the first place.

Here we can see all 3 forms:
Luke 11:13, Christ Jesus, our God:

If you then, though you are evil, know how to give good gifts to your
children, how much more will your Father in heaven give the Holy
Spirit to those who ask him!

So the Holy Spirit can dwell inside Man (Acts 10:44). But not The Son nor The Father. Not all 3 forms dwelling in man like you proposed.

Sin is something our God, The Father wants to pass an eye over and address accordingly thru the Holy Spirit. Because He loves us. He came to save humanity.